Beyond ANOVA (Part 7.2) {Fundamentals of RMA}

[Neeraj Kaushik]

Fundamentals of ANOVA (re-visited):
 
Lets revise ANOVA fundamental by taking the previous example of car.sav file. Here we've Sales for the car, jeep & trucks.
 
Lets start with the fundamental question.
Why are there difference in the sales figures of cars? We can understand that difference in the sales figures of car & jeep may be attributed coz of differences in vehicle, but for same category of cars why sales differ??
The answer is simple that these differences are by chance. Since in nature we've variations, so its natural to see these differences even when the same thing (e.g. sales) is measured in same category (e.g. car)
 
Now the 2nd question: Why there are differences between the Sales of car, jeep & truck?
One answer is again By chance and 2nd answer is coz the 3 categories are different. So which one we shd believe?
So what we do is lets take a ratio of Between categories variation to Within category variations. If this ratio comes less than the table-value of F-test then the differences are by chance else by the difference in the 3 categories.
 
What is the meaning of Table value?
Lets say F value is 3, it means we expect the variation between samples to be 3 times that of within sample variations. So if the calculated ratio comes to be 2.5 then it means the difference between the samples is coz of chance factor.
So if calculated F values comes 10, then it can't be coz of chance factor ans indeed the categories that we are comparing differ significantly.
 
Now we can very well understand why there was we has assumption of homogenity of variance as if variance between the samples is not same then automatically we'll get the biased value of F and there'll be spurious results. 
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Fundamentals of RMA:
 
Now when we've RMA, there's no between sample variations coz here all readings are of the same respondent. So here we'll assume the total variations are the sum of within sample variations (chance factor) and the variations coz the experimental conditions (different time periods) are not the same.
 
So the first thing is that there's no such assumption of homogenity of variance. Actually its there in somewhat other form, which is called Sphericity here. 
Sphericity refers to the equality of variances of the differences between treatment levels. So, if we were to take each pair of treatment levels, and calculate the differences between each pair of scores, then it is necessary that these differences have approximately equal variances.
 
In SPSS it is called as Test for the Sphericity and the Statistics was given by Mauchly. So while reading the output of RMA, the first thing we'll look for Mauchly's test of Sphericity.
 
Now if p-value for this test is < 0.05 it means assumption is violated. But the good thing here is that pvalue < 0.05 wont make this  RMA infeasible rather SPSS will give us 2 correction factors namely Greenhouse-Geisser and Huynh-Feldt Corrections (called Epsilon here).
One important thing to note here is that these corrections donot change the statistics values, they just correct the degree of freedom hence F-value & p-value changes.
 
Rules for using corrections:
1. If p-value for Mauchly's test of Spherricity > 0.05 Go to the next table of Test of within subjects effects and interpret from the row of Sphericity assumed.
Else next step.     
2. If p-value for Mauchly's test of Spherricity < 0.05, see the value of Greenhouse-Geisser. If this value is < 0.75 then go to the next table of within subjects effects and interpret from the row of Greenhouse-Geisser. 
Else next step.     
3. The value of Greenhouse-Geisser varies between value given by lower bound (last column of this table) and 1.
When this value is >0.75 then it start giving biased results, hence then we'll use the 2nd correction of Huynh-Feldt Corrections. So we'll go to the next table of within subjects effects and interpret from the 
row of Huynh-Feldt. 
4. When the assumption of sphericity is violated, we'll also check the first table of Multivariate test. At times we may have complete different results from the 2 corrections, then this table will make sense.
 
Rest in next post
 
Happy Learning
Neeraj